Exact off-shell Sudakov form factor in N=4 SYM

We consider the Sudakov form factor in N=4 SYM in the off-shell kinematical regime, which can be achieved by considering the theory on its Coulomb branch. We demonstrate that up two three loops both the infrared-divergent as well as the finite terms do exponentiate, with the coefficient accompanying $\log^2(m^2)$ determined by the octagon anomalous dimension $\Gamma_{oct}$. This behaviour is in strike contrast to previous conjectural accounts in the literature. Together with the finite terms we observe that up to three loops the logarithm of the Sudakov form factor is identical to twice the logarithm of the null octagon $\mathbb{O}_0$, which was recently introduced within the context of integrability-based approaches to four point correlation functions with infinitely-large R-charges. The null octagon $\mathbb{O}_0$ is known in a closed form for all values of the 't Hooft coupling constant and kinematical parameters. We conjecture that the relation between $\mathbb{O}_0$ and the off-shell Sudakov form factor will hold to all loop orders.